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17
The quantitative structure of exponential time
- Complexity Theory Retrospective II
, 1997
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Equivalence of Measures of Complexity Classes
"... The resource-bounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomial-time computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases ..."
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Cited by 73 (22 self)
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The resource-bounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomial-time computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases) fi i 2 [ffi; 1 \Gamma ffi], and any complexity class C (such as P, NP, BPP, P/Poly, PH, PSPACE, etc.) that is closed under positive, polynomial-time, truth-table reductions with queries of at most linear length, it is shown that the following two conditions are equivalent. (1) C has p-measure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the coin-toss probability measure given by the sequence ~ fi. (2) C has p-measure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the uniform probability measure. The proof introduces three techniques that may be useful in other contexts, namely, (i) the transformation of an efficient martingale for one probability measu...
The Complexity and Distribution of Hard Problems
- SIAM JOURNAL ON COMPUTING
, 1993
"... Measure-theoretic aspects of the P m -reducibility structure of the exponential time complexity classes E=DTIME(2 linear ) and E 2 = DTIME(2 polynomial ) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in ..."
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Cited by 49 (18 self)
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Measure-theoretic aspects of the P m -reducibility structure of the exponential time complexity classes E=DTIME(2 linear ) and E 2 = DTIME(2 polynomial ) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are P m - hard for E and other complexity classes. Tight upper and lower bounds on the size of complexity cores of hard languages are derived. The upper bound says that the P m -hard languages for E are unusually simple, in the sense that they have smaller complexity cores than most languages in E. It follows that the P m -complete languages for E form a measure 0 subset of E (and similarly in E 2 ). This latter fact is seen to be a special case of a more general theorem, namely, that every P m -degree (e.g., the degree of all P m -complete languages for NP) has measure 0 in E and in E 2 .
Resource-Bounded Measure and Randomness
"... We survey recent results on resource-bounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes and . Moreover, we treat time-bounded genericity and stochasticity concepts which are weaker than ..."
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Cited by 42 (6 self)
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We survey recent results on resource-bounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes and . Moreover, we treat time-bounded genericity and stochasticity concepts which are weaker than time-bounded randomness but which suffice for many of the applications in complexity theory.
Resource Bounded Randomness and Weakly Complete Problems
- Theoretical Computer Science
, 1994
"... We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure ([7, 8]). We concentrate on n c - randomness (c 2) which corresponds to the polynomial time bounded (p-) measure of Lutz, and which is adequate for studying the internal and quantitative s ..."
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Cited by 36 (7 self)
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We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure ([7, 8]). We concentrate on n c - randomness (c 2) which corresponds to the polynomial time bounded (p-) measure of Lutz, and which is adequate for studying the internal and quantitative structure of E = DTIME(2 lin ). However we will also comment on E2 = DTIME(2 pol ) and its corresponding (p2 -) measure. First we show that the class of n c -random sets has p-measure 1. This provides a new, simplified approach to p-measure 1-results. Next we compare randomness with genericity (in the sense of [2, 3]) and we show that n c+1 - random sets are n c -generic, whereas the converse fails. From the former we conclude that n c -random sets are not p-btt-complete for E. Our technical main results describe the distribution of the n c -random sets under p-m-reducibility. We show that every n c -random set in E has n k -random predecessors in E for any k 1, whereas the amou...
Resource-Bounded Baire Category: A Stronger Approach
- Proceedings of the Tenth Annual IEEE Conference on Structure in Complexity Theory
, 1996
"... This paper introduces a new definition of resource-bounded Baire category in the style of Lutz. This definition gives an almost-all/almost-none theory of various complexity classes. The meagerness/comeagerness of many more classes can be resolved in the new definition than in previous definitions. ..."
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Cited by 8 (0 self)
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This paper introduces a new definition of resource-bounded Baire category in the style of Lutz. This definition gives an almost-all/almost-none theory of various complexity classes. The meagerness/comeagerness of many more classes can be resolved in the new definition than in previous definitions. For example, almost no sets in EXP are EXP-complete, and NP is PF-meager unless NP = EXP. It is also seen under the new definition that no rec-random set can be (recursively) tt-reducible to any PF-generic set. We weaken our definition by putting arbitrary bounds on the length of extension strategies, obtaining a spectrum of different theories of Baire Category that includes Lutz's original definition. 1
Weakly Useful Sequences
, 2004
"... An infinite binary sequence x is defined to be (i) strongly useful if there is a computable time bound within which every decidable sequence is Turing reducible to x; and (ii) weakly useful if there is a computable time bound within which all the sequences in a non-measure 0 subset of the set of dec ..."
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Cited by 7 (2 self)
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An infinite binary sequence x is defined to be (i) strongly useful if there is a computable time bound within which every decidable sequence is Turing reducible to x; and (ii) weakly useful if there is a computable time bound within which all the sequences in a non-measure 0 subset of the set of decidable sequences are Turing reducible to x. Juedes,
Almost Complete Sets
, 2000
"... . We show that there is a set which is almost complete but not complete under polynomial-time many-one (p-m) reductions for the class E of sets computable in deterministic time 2 lin . Here a set A in a complexity class C is almost complete for C under some reducibility r if the class of the p ..."
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Cited by 3 (2 self)
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. We show that there is a set which is almost complete but not complete under polynomial-time many-one (p-m) reductions for the class E of sets computable in deterministic time 2 lin . Here a set A in a complexity class C is almost complete for C under some reducibility r if the class of the problems in C which do not r-reduce to A has measure 0 in C in the sense of Lutz's resource-bounded measure theory. We also show that the almost complete sets for E under polynomial-time bounded one-one length-increasing reductions and truth-table reductions of norm 1 coincide with the almost p-m-complete sets for E. Moreover, we obtain similar results for the class EXP of sets computable in deterministic time 2 poly . 1 Introduction Lutz [15] introduced measure concepts for the standard deterministic time and space complexity classes which contain the class E of sets computable in deterministic time 2 lin . These measure concepts have been used for investigating quantitative aspe...
